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Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these theorems do?

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Yes, there are constructive proofs for fixed point theorems including Brouwer. Also, the proof of the Banach fixed point theorem with which I am most familiar is constructive. In fact, here is a paper all about constructive methods for fixed point theorems:

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+1: Nice reference. –  copper.hat Jul 24 at 7:08
    
So that means we can find an algorithm for computing a fixed point? (for irrational numbers, standard interpretation of computing "reals" apply) –  Kalami Jul 24 at 13:37

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